factorion

where d1 is the least significant digit and dk is the most significant, if it is also the case that

n=∑i=1kdi!

then n is a factorion. In other words, the sum of the factorials of the digits in a standard positional integer base b (such as base 10) gives the same number as multiplying the digits by the appropriate power of that base. With the exception of 1, the factorial base representation of a factorion is always different from that in the integer base. Obviously, all numbers are factorions in factorial base.

1 is a factorion in any integer base. 2 is a factorion in all integer bases except binary. In base 10, there are only four factorions: 1, 2, 145 and 40585. For example, 4×104+0×103+5×102+8×101+5×100=4!+0!+5!+8!+5!=40585. (The factorial base representation of 40585 is 10021001).